(a) Suppose that the graph of f (x) is above the x-axis and concave down on the interval a₀ ≤ x ≤ a₁. Let x₁ be the midpoint of this interval, let Δx = a₁ - a₀, and construct the line tangent to the graph of f (x) at (x₁, f (x₁)), as in Fig. 15(a). Show that the area of the shaded trapezoid in Fig. 15(a) is the same as the area of the shaded rectangle in Fig. 15(c), that is, f (x₁)Δx.

(b) Suppose that the graph of f (x) is above the x-axis and concave down for all x in the interval a ≤ x ≤ b. Explain why T ≤ ∫_a^b f (x)dx ≤ M, where T and M are the approximations given by the trapezoidal and midpoint rules, respectively.

Transcribed Image Text:

(x1, f(x1)) do X1 (a) Figure 15 y = f(x) a1 do Δ.χ. X1 (b) a1 do y = f(x) Δ.χ. + X1 (c) a1